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In this paper, we propose a discrete SIR epidemic model whose discretization scheme preserves the global stability of equilibria for a class of continuous SIR epidemic models. From a biological motivation, the infection rate of the model is given by unspecified functions which incorporates a latency period with some distribution. By identifying the basic reproduction number R 0 of the model, the global asymptotic stability of the equilibria of the model is fully determined by applying discrete-time analogue of Lyapunov functionals when the infection rate has a suitable monotone property. Moreover, our result indicates that the latency period does not influence the global dynamics of the model.

In this paper, we consider the following logistic equation with piecewise constant arguments:where r > 0, a 0 , a 1 , . . . , a m 0, m j =0 a j > 0, and [x] means the maximal integer not greater than x. The sequence {N n } ∞ n=0 , where N n = N(n), n = 0, 1, 2, . . . , satisfies the difference equationUnder the condition that the first term a 0 dominates the other m coefficients a i , 1 i m, we establish new sufficient conditions of the global asymptotic stability for the positive equilibrium N * = 1/( m j =0 a j ).

Keywords: SIRS epidemic model Nonlinear incidence rate Global asymptotic stability Permanence Monotone iterative technique Lyapunov functional technique In this paper, we investigate a disease transmission model of SIRS type with latent period τ 0 and the specific nonmonotone incidence rate, namely,.For the basic reproduction number R 0 > 1, applying monotone iterative techniques, we establish sufficient conditions for the global asymptotic stability of endemic equilibrium of system which become partial answers to the open problem in [Hai-Feng Huo, ZhanPing Ma, Dynamics of a delayed epidemic model with non-monotonic incidence rate, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 459-468]. Moreover, combining both monotone iterative techniques and the Lyapunov functional techniques to an SIR model by perturbation, we derive another type of sufficient conditions for the global asymptotic stability of the endemic equilibrium.

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In this paper, we establish the global asymptotic stability of equilibria for an SIR model of infectious diseases with distributed time delays governed by a wide class of nonlinear incidence rates. We obtain the global properties of the model by proving the permanence and constructing a suitable Lyapunov functional. Under some suitable assumptions on the nonlinear term in the incidence rate, the global dynamics of the model is completely determined by the basic reproduction number R 0 and the distributed delays do not influence the global dynamics of the model.